1-1 Motivation
The study of fluid flow in microchannels is of significant interest to engineers because of microfluid have wide industrial applications. For examples, miniaturize flow injection analysis, micro-reactors for the analysis of biological cells, heat sinks for the analysis of biological cells, and heat sinks for cooling microchips and laser diode arrays, etc.
At the small scales channels, the electrical double layer (EDL) phenomena significantly makes an influence on the fluid behaviors and the electrical double layer is also the important interfacial effect.
The EDL field in such a microchannels is governed by the Nonlinear Poisson-Boltzmann (PB) equation.
The Nonlinear Poisson-Boltzmann equation is also used for the description of the distribution of electrostatic potential in colloidal dispersions. Knowing the electrostatic potential, one can calculate other quantities such as the force of particle-particle interaction. Features of particle interaction are great importance for the stability and properties of colloidal dispersions. Numerical investigation of models based on the PB equation can provide important information on effective particle interaction in colloidal systems.
In previous documents and theses, there are not so many reports discussed the three-dimensional computation model. The two-dimensional computation model was most commonly mentioned. However, when the geometries is getting complex, the two-dimensional computation model is gradually losing its accuracy.
Therefore, we have developed a numerical scheme to simulate Electrical double layer potential distribution by solving 3D Nonlinear Poisson-Boltzmann equation.
1-2 Background
1-2.1 The Like-charge Attractions Phenomenon
Colloidal sphere provide a simple model system for understanding the interactions of the charge particles in a salt solution. Hence, it came as a great surprise when it was observed that two like-charge spheres can attract each other when the spheres are confined by walls. Since both the charge densities and sizes of the spheres are in the range of large proteins, it would be expected that a charge in sign of this interaction would have important implications for biological systems.
The attractive interaction between two charge walls can be understood with a simple picture (Fig 1.1). When the spheres are sufficiently close to the wall, they are electrostatically repelled from it. The net force on each sphere thus includes both their mutual electrostatic repulsion and their repulsion from the wall. How the spheres respond depends on their hydrodynamic mobility: when the spheres are close together, their mutual repulsion overwhelms any hydrodynamic coupling, and the spheres will separate as hydrodynamic coupling; the spheres will separate as expected for like-charge bodies. However, when they are beyond some critical separation, the hydrodynamic coupling due to the wall force overcomes the electrostatic repulsion, so that the relative distance between the spheres decreases as they move away from the wall.
1-2.2 Electric double layer
Electrokinetic phenomena are of considerable importance in many fields of science and engineering. In particular, they exert a strong influence on the flow behavior of a fluid in microchannels and capillaries. Most solid surfaces bear electrostatic charges. When a charge surface is in contact with an electrolyte, the
electrostatic charges on the solid surface will influence the distribution of nearby ions in the electrolyte solution. Ions of opposite of charges to that of the surface are attracted towards the surface, while ions of like charges are repelled from the surface;
thus, an electric field is established. The charges on the solid surface and the balancing charge in the liquid are called the “electric double layer” (EDL). The EDL potential distribution is show in Fig 1.2. The sign and magnitude of the EDL field depend on the nature of the surface and the liquid.
The distribution of EDL can be class with two regions. One is compact or stern layer the other is diffusion layer. Guoy and Chapmal modeled the region near the surface as a diffuse double layer, where they linked the nonuniform ion distribution to the competing electrical and thermal diffusion forces [1]. Stern later presented the basis for the current model, in which the Stern plane splits the EDL into an inner, compact layer and an outer, diffuse layer. In the inner layer or the Stern layer, the geometry of the ions and molecules strongly influences the charge and potential distribution, with the Stern plane located near the surface, at roughly the radius of a hydrated ion. The inner layer between the surface and the Stern plane is considered to be immobile; if the ions are within the Stern plane, thermal diffusion will not be strong enough to overcome electrostatic or Van der Waals forces and they will attach to the surface, becoming specifically adsorbed [1]. In the outer diffuse layer, the ions are far enough away from the surface that they are mobile. Electrokinetic transport phenomena such as electroosmosis can be understood in terms of the surface potential at the surface of shear, because these phenomena are only directly related to the mobile part of the EDL [1].
Within the diffuse layer, because of the EDL, the net charge density ρe is not zero. If an electric field is applied along the length of the channel, a body force is
exerted on the ions in the diffuse layer of the EDL. The ions will move under the influence of the applied electrical field, pulling the liquid with them and resulting in electroosmotic flow. The fluid movement is carried through to the rest of the fluid in the channel by viscous forces. This electrokinetic process is called electroomosis.
1-3 Literature Survey
The EDL field is described by the Poisson-Boltzmann equation, which is a three-dimensional, nonlinear, second-order partial differential equation. The hyperbolic sine term makes the second-order different equation exponentially nonlinear; therefore. A general analytical solution is not possible.
The PB equation can model many phenomenon such as the like-charge interaction, electrokinetic flow and micro-fluidic actuation etc. There are many studies in the literature dealing with the solution of Poisson-Boltzmann equation.
Bowen and Sharif [2] they presented a 2D FE solution combined with adaptive mesh refinement. They used the Debye-Huckel solution to initial guess, and the Newton sequence was used to solve the nonlinear hyperbolic sine term. They used nine-node quadrilateral elements. Bowen and Sharif also presented some results in 1998 journal of Nature [3]. They said their solver can model the like-charge attractions phenomenon which is presented in 1997 journal of Nature by Larsen and Grier [4]. Dyshlovenko [5] also presented a 2D FE solution combined with adaptive mesh refinement but they used six-node triangular elements. In 2002, Dyshlovenko publish their result in CPC [6], but Dyshlovenko is different from Bowen and Sharif, Dyshlovenko do not find the like-charge attractions phenomenon in his results.
Tuinier [7] proposed the approximate solutions to the PB equation in spherical and cylindrical. The approximate solutions provided me a good initial guess for my PB
solver. Das and Bhattacharjee [8] examine the results in 2004, they presented finite element simulation results which is considered the roughness on the wall. They think roughness on the wall can influence the electrostatic forces and the like-charge attraction.
1.4 Objectives and Organization of the Thesis
Based on previous reviews, the current objectives of the thesis are summarized as follows:
1. To develop a parallel 3-D finite-element solution of the nonlinear Poisson-Boltzmann equation.
2. To couple the parallel adaptive mesh refinement with this PB solver.
The organization of this thesis would be stated as follow: First is this introduction, followed by the numerical method. Next would be the Results and discussions, and finally the Conclusions and Future work.